Theorem xmstopn 24508

Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
isms.j	⊢ 𝐽 = (TopOpen‘𝐾)
isms.x	⊢ 𝑋 = (Base‘𝐾)
isms.d	⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
xmstopn	⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn

Step	Hyp	Ref	Expression
1		isms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
2		isms.x	. . 3 ⊢ 𝑋 = (Base‘𝐾)
3		isms.d	. . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
4	1, 2, 3	isxms 24504	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
5	4	simprbi 501	1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142   × cxp 5645   ↾ cres 5649  ‘cfv 6521  Basecbs 17245  distcds 17295  TopOpenctopn 17450  MetOpencmopn 21411  TopSpctps 22989  ∞MetSpcxms 24374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-res 5659  df-iota 6477  df-fv 6529  df-xms 24377

This theorem is referenced by:  imasf1oxms  24546  ressxms  24582  prdsxmslem2  24586  tmsxpsmopn  24594  xmsusp  24626  cmetcusp1  25412  minveclem4a  25489  minveclem4  25491  qqhcn  34285  rrhcn  34291  rrexthaus  34301  dya2icoseg2  34572  sitmcl  34645